Patterns and Sequences and Formulas Part 2 With Clarificatio

[JohnT1111]

Alex draws a square and shades four corners. One side of each new square is one quarter the length of the original square. Using the corners fo the shaded squares, Alex draws another square, and then shades its four corners in the same manner. If she repeats this pattern over and over again, what portion of the orginal square will be shaded?

Eliz. asked me about clarification..but this is the entire question. For visualization: it starts with one big square which has four corners shaded. Then another square is formed inside the big square, with all its corners at the edge of the shaded-squares. As so on and on.

I have to write a formula and a sequence to model it basically.

I figured it out this far: this shaded corners on teh first square take up 1/4 of its total area, and the next square 1/16 of the total area. As a sequence:
4^-1 , 4 ^-2, 4^-3, ... etc. ^ represents exponent.

Anyway I need a formula to model that. It can be recursive, arithmetic, geometric..no matter. Tn = 4n^-1 for example.

Sorry if i havent clarified enough, if anyone could help..

 

[galactus]

Apparently I misinterpreted the problem. Sorry

Apparently, I misinterpreted minsinterpreting the problem.

\(\displaystyle \sum_{n=1}^{\infty}(\frac{1}{4^{n}})=\frac{a_{1}}{1-r}\)

\(\displaystyle a_{1}\) is the first term, \(\displaystyle \frac{1}{4}\) and r is the

common ratio \(\displaystyle \frac{1}{4}\).

\(\displaystyle \frac{\frac{1}{4}}{1-\frac{1}{4}}=\frac{1}{3}\)

 

[stapel]

Re: Patterns and Sequences and Formulas Part 2 With Clarific

Alex draws a square and shades four corners. One side of each new square is one quarter the length of the original square. Using the corners fo the shaded squares, Alex draws another square, and then shades its four corners in the same manner. If she repeats this pattern over and over again, what portion of the orginal square will be shaded?

For visualization: it starts with one big square which has four corners shaded. Then another square is formed inside the big square, with all its corners at the edge of the shaded-squares. As so on and on.

This is how I'm reading the exercise:

There is an original square. Think of the square as being divided into a four-by-four grid. Shade the corner squares of the grid, leaving something vaguely similar to a really fat "plus" sign.

Do you work only with the shaded part, or are the shaded parts discarded? Are the "new" squares the old corners, or something else? (If the corners are fully shaded, and the "new" squares are the corners, then there is nothing to "add" to the shaded portion, since it's fully shaded.)

Thank you.

Eliz.

 

[galactus]

Wrong drawing!

 

[Denis]

Re: Patterns and Sequences and Formulas Part 2 With Clarific

Eliz. asked me about clarification..but this is the entire question. For visualization: it starts with one big square which has four corners shaded.

C'mon John:
...which has FOUR SHADED SQUARES, ONE IN EACH CORNER :evil:

Anyway you got it here: http://www.sosmath.com/CBB/viewtopic.php?t=21090